This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present technological advancement. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present technological advancement. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
The perfectly matched layers (PML) absorbing boundary condition by Berenger (1994) is commonly used to approximate the radiation boundary condition for the sides and bottom of an earth model where the earth model is assumed to have infinite extent but the computational model has finite extent. Waves should not reflect from external boundaries of the computational model that are designated to have the radiation boundary condition.
In the standard form for PML as described by Marcinkovich and Olsen (2003), every derivative normal to an exterior boundary has a wave field dissipation operator applied. Several issues arise with standard PML operators. For general anisotropy, if the group velocity and the phase velocity have different signs for the direction normal to the boundary, PML goes unstable and energy can be amplified rather than attenuated at the boundary. The conventional design of PML dissipates in the direction of phase velocity, not in the actual direction of energy propagation as will be discussed below in the detailed description section. In addition, for elastic or anisotropic elastic wave propagation, thin high-contrast shear-velocity velocity layers on the boundary can create instability due to boundary or interface waves not behaving like body waves and again having a phase velocity vector with a different sign compared to the group velocity vector component normal to the boundary.
An ad hoc fix to the thin high-contrast shear wave velocity anomaly in the boundary zone has been to smooth the shear velocity earth model in the PML boundary zone. The smoother on the shear wave velocity needs to honor the rock physics constraints for stability and can be sensitive to how a full waveform inversion updates parameters near the boundary.
The frequency-domain form of the PML operator (eqn. 1)
                                                        (                                                ∂                                                                                                          ∂                                      x                    l                                                              )                        PML                    ⁢                      F            ⁡                          (                              x                →                            )                                      =                              (                          1                              1                +                                                                            ω                      l                                        ⁡                                          (                                              x                        →                                            )                                                                            i                    ⁢                                                                                  ⁢                    ω                                                                        )                    ⁢                      (                                          ∂                                                                                              ∂                                  x                  l                                                      )                    ⁢                      F            ⁡                          (                              x                →                            )                                                          (        1        )            and the non-split PML (NPML) operator (eqn. 2)
                                                        (                                                ∂                                                                                                          ∂                                      x                    l                                                              )                        NPML                    ⁢                      F            ⁡                          (                              x                →                            )                                      =                              (                                          ∂                                                                                              ∂                                  x                  l                                                      )                    ⁢                      (                          1                              1                +                                                                            ω                      l                                        ⁡                                          (                                              x                        →                                            )                                                                            i                    ⁢                                                                                  ⁢                    ω                                                                        )                    ⁢                      F            ⁡                          (                              x                →                            )                                                          (        2        )            
to be associated with spatial derivative terms are given above. The NPML can be easier to implement, but the results may not be as good as those achieved with the PML. To mitigate reflections from an external earth model boundary, derivative terms in the original set of wave propagation equations are replaced with either PML or NPML derivatives which damp the waves propagating to and from the boundary. In three-dimensional space, l=1, 2 or 3, and the above formulation allows the frequency in a direction normal to the boundary, which may be called the damping frequency parameter to be different from the damping frequency parameter in the other two directions. In the time domain, these operators correspond to a temporal convolution with a damped exponential in time. The difference between the PML operator and the NPML operator is an exchange of the order of the spatial derivative and the dissipation operator. These two operators are transposes of each other. They are not identical because the damping coefficients are spatially dependent. The dissipation operator Di can accordingly be defined in the frequency domain as
                                          D            l                    ⁡                      (                          ω              ,                              x                →                                      )                          =                  (                      1                          1              +                                                                    ω                    l                                    ⁡                                      (                                          x                      →                                        )                                                                    i                  ⁢                                                                          ⁢                  ω                                                              )                                    (        3        )            where ωi may be called the damping frequency parameter and ω is the frequency of propagation.